with a 120° angle and having integer sides in arithmetic progression. The only integer triangle with area = semiperimeter has sides (3, 4, 5). The only integer triangles with area = perimeter have sides (5, 12, 13), (6, 8, 10), (6, 25, 29), (7, 15, 20), and (9, 10, 17). Of these the first two, but not the last three, are right triangles. There exist integer triangles with three rational medians. The smallest has sides (68, 85, 87). Others include (127, 131, 158), (113, 243, 290), (145, 207, 328) and (327, 386, 409). There are no isosceles Pythagorean triangles. The only primitive Pythagorean triangles for which the square of the perimeter equals an integer multiple of the area are (3, 4, 5) with perimeter 12 and area 6 and with the ratio of perimeter squared to area being 24; (5, 12, 13) with perimeter 30 and area 30 and with the ratio of perimeter squared to area being 30; and (9, 40, 41) with perimeter 90 and area 180 and with the ratio of perimeter squared to area being 45. There exists a unique (up to similitude) pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. The unique pair consists of the (377, 135, 352) triangle and the (366, 366, 132) triangle. There is no pair of such triangles if the triangles are also required to be primitive integral triangles. The authors stress the striking fact that the second assertion can be proved by an elementary argumentation (they do so in their appendix A), whilst the first assertion needs modern highly non-trivial mathematics. See also Robbins pentagon, a cyclic pentagon with integer sides and integer area Euler brick, a cuboid with integer edges and integer face diagonals Tetrahedron#Integer tetrahedra References Arithmetic problems of plane geometry Discrete geometry Squares in number theory Types of triangles